Let be a regular local (noetherian) ring of dimension . In the previous post, we described loosely the local cohomology functors
(in fact, described them in three different ways), and proved a fundamental duality theorem
Here is the Matlis duality functor , for an injective envelope of the residue field . This was stated initially as a result in the derived category, but we are going to use the above form.
The duality can be rewritten in a manner analogous to Serre duality. We have that (in fact, this could be taken as a definition of ). For any , there is a Yoneda pairing
and the local duality theorem states that it is a perfect pairing.
Example 1 Let be an algebraically closed field, and suppose that is the local ring of a closed point on a smooth -variety . Then we can take for the module
in other words, the module of -linear distributions (supported at that point). To see this, note that defines a duality functor on the category of finite length -modules, and any such duality functor is unique. The associated representing object for this duality functor is precisely .
In this case, we can think intuitively of as the cohomology
These can be represented by meromorphic -forms defined near ; any such defines a distribution by sending a function defined near to . I’m not sure to what extent one can write an actual comparison theorem with the complex case. (more…)